Journal of Modern Dynamics (JMD)

Ergodic infinite group extensions of geodesic flows on translation surfaces

Pages: 477 - 497, Issue 4, October 2012      doi:10.3934/jmd.2012.6.477

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David Ralston - SUNY College at Old Westbury, Mathematics/CIS Department, P.O. Box 210, Old Westbury, NY 11568, United States (email)
Serge Troubetzkoy - Aix-Marseille University, CNRS, CPT, IML, Frumam, 13288 Marseille Cedex 09, France (email)

Abstract: We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. K. Frączek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases.

Keywords:  Translation surface, ergodicity, diophantine approximation, essential values.
Mathematics Subject Classification:  Primary: 37A25, 37A40; Secondary: 37A20, 37C40.

Received: May 2012;      Available Online: January 2013.